Generating Heavy-Tailed Financial Returns via Elliptical Variational Auto-Encoders

Financial risk management often involves learning the return tail distributions, which include both market co-movement and asset-specific extremes. Inspired by the elliptical approximate factor model in the financial literature, we develop an elliptical variational auto-encoder (E-VAE) based on its nonlinear extension, which decomposes financial returns into a nonlinear common factor component and idiosyncratic fluctuations; both follow heavy-tailed distributions with occasional jumps. The elliptical VAE admits a standard evidence lower bound with analytic Kullback–Leibler terms and reparameterized training. In financial applications, the fitted encoder provides posterior estimates of the common factors from historical observations, while the fitted decoder can generate synthetic return vectors for downstream risk assessment and portfolio planning tasks. By exploiting the low-dimensional factor structure, our model enables efficient sampling of high-dimensional return vectors. Experiments on synthetic data and the U.S. and Chinese stock markets demonstrate its ability to capture both the body and tails of return distributions, supporting its use in downstream financial decision-making.